What is the ruling regarding my four-year-old daughter swimming in the pool in a " شرعي" (Islamic compliant) swimsuit, given my husband's objection to it?

You must obey your husband in matters that do not contradict Islamic law. And know that the 'awrah (private parts) of a boy or girl under seven years of age has no ruling. It is recommended to accustom children to cover their 'awrah from a young age, and also to accustom a girl to avoid mixing with males in their play areas.